52 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA freely on ?r(E), that for each z e E the orbit R * TT(Z) is closed in ?r(E), that the canonical mapping R ->R • TT(Z) is a homeomorphism, and that the orbit space 7r(E)/R is separ- able, metrizable and locally compact; but that R does not act properly on 7r(E). 4. In R3, let E be the union of the sets Cfl x {z} for 0 ^> 1 and z ^ 0 (notation of Problem 2) and the line DO and the lines Dz for z> 0, where DO is the set of points (t, — 1, 0) with / eR, and D" is the set of points (t, 1, z) with t e R. The additive group R acts continuously on E as follows: (1) s • (t, —1, 0) = (s + t, -1, 0); (2) ,y • (fa(t), z) = (fa(s + 0, *); (3) s-(t9\9z) = (t — s9l9 z). Show that the orbits of this action have the same properties as in Problems 2 and 3, and that the orbit space E/R is Hausdorff but not metrizable. 5. Let E be a locally compact metrizable space and let G be a topological group acting continuously on E. Let TT : E -> E/G be the canonical map. Suppose that E/G is Hausdorff. (a) Let K be a compact subset of E and let U be an open neighborhood of K. Show that there exists a continuous mapping of E into [0,1] which takes the value 1 on 7r~1(7r(K)) and the value 0 on the complement of 7r"1(rr(U)). (b) Deduce from (a) that there exists a continuous mapping of E/G into [0,1], taking the value 1 on 7r(K) and the value 0 on the complement of 7r(U). (Show first that there exists a relatively compact open neighborhood Ui of K in E such that 0i c U, Deduce that there exist two continuous mappings /i,/2 of E into [0, J] such that/! takes the value J on 7r'1(n(Ky) and the value 0 on E —7r-J(7r(Ui)), and/2 takes the value J on 7r~1(7r(O1)) and the value 0 on E —Tr'^TrCU)). Consider the function/i -\~f2. Iterate this "interpolation" indefinitely and pass to the limit.) (c) If E is separable, show that there exists a sequence (Un) of relatively compact open sets in E such that the 7r(Un) form a basis for the topology of E/G. (d) Suppose that E is separable. Show that E/G is metrizable. (For each pair of indices m9 n such that Om <= UB, consider a continuous mapping fmn of E/G into [0,1] which takes the value 1 on 7r(Um) and the value 0 on the complement of 7r(Un). Consider the continuous mapping x\—> (/„«(*)) of E/G into the product space RN x N). 6. If M is a monoid with neutral element e, we define an action of M on a set E as at the beginning of (12.10). Suppose that E is a compact metric space and that, for each s e M, the mapping XY-+S • x is continuous. The closed orbit ofx with respect to M is defined to be the set M • x\ it is stable under the action of M. Show that for each x e E there exists in M • x a minimal closed orbit Z (i.e., such that M • z = Z for all z e Z). (For each y e M • x let X(y) be the least upper bound, as t runs through M * y9 of the distances from / to a closed orbit contained in M • t. Show that the greatest lower bound of the numbers A(y) with y e M • x is 0, by showing that otherwise E would not be precompact. Deduce that there exists a sequence (yn) of points of M • x such that yn+1 e M • yn and the sequence (X(yn)) tends to 0. Now use the compactness of E to complete the proof.) 11. HOMOGENEOUS SPACES Let G be a group, H a subgroup of G. Recall that the set of left cosets xH (resp. right cosets Rx) of H in G is denoted by G/H (resp. H\G). For each 1] into G is continuous. Deduce that this mapping can be extended to a non-